Structural Misspecification Research Articles

Path and Direction Discovery in Individual Dynamic Factor Models: A Regularized Hybrid Unified Structural Equation Modeling with Latent Variable

There has been an increasing call to model multivariate time series data with measurement error. The combination of latent factors with a vector autoregressive (VAR) model leads to the dynamic factor model (DFM), in which dynamic relations are derived within factor series, among factors and observed time series, or both. However, a few limitations exist in the current DFM representatives and estimation: (1) the dynamic component contains either directed or undirected contemporaneous relations, but not both, (2) selecting the optimal model in exploratory DFM is a challenge, (3) the consequences of structural misspecifications from model selection is barely studied. Our paper serves to advance DFM with a hybrid VAR representations and the utilization of LASSO regularization to select dynamic implied instrumental variable, two-stage least squares (MIIV-2SLS) estimation. Our proposed method highlights the flexibility in modeling the directions of dynamic relations with a robust estimation. We aim to offer researchers guidance on model selection and estimation in person-centered dynamic assessments.

The Sensitivity of Bayesian Fit Indices to Structural Misspecification in Structural Equation Modeling

This study examined the false positive (FP) rates and sensitivity of Bayesian fit indices to structural misspecification in Bayesian structural equation modeling. The impact of measurement quality, sample size, model size, the magnitude of misspecified path effect, and the choice or prior on the performance of the fit indices was also investigated. The Bayesian fit indices examined in this study included PPP, DIC, BRMSEA, BCFI, BTLI. The results from the simulation study showed that BRMSEA, BCFI, and BTLI failed to detect structural misspecification. The performance of DIC depended majorly on measurement quality, and the sensitivity of PPP depended on sample size, measurement quality, the magnitude of the omitted path effect, and the choice of prior. Informative prior with too narrow precisions resulted in higher FP rates. Empirical implications for applied researchers and future research directions were discussed.

Misspecification and unreliable interpretations in psychology and social science.

The replicability crisis has drawn attention to numerous weaknesses in psychology and social science research practice. In this work we focus on three issues that cannot be addressed with replication alone, and which deserve more attention: Functional misspecification, structural misspecification, and unreliable interpretation of results. We demonstrate a number of possible consequences via simulation, and provide recommendations for researchers to improve their research practice. Psychologists and social scientists should engage with these areas of analytical and statistical improvement, as they have the potential to seriously hinder scientific progress. Every research question and hypothesis may present its own unique challenges, and it is only through an awareness and understanding of varied statistical methods for predictive and causal modeling, that researchers will have the tools with which to appropriately address them. (PsycInfo Database Record (c) 2021 APA, all rights reserved).

Revisiting the Impact of Measurement Quality on Targeted Structural Model Fit Indexes in Structural Equation Modeling

This study examined the false positive rates and sensitivity of structural fit indexes to structural misspecification in SEM. The impact of sample size, model size, the magnitude of standardized factor loadings, and misspecified path effect on the fit indexes was also investigated. The structural fit indexes examined in this study included RMSEA-P, C9, C10 recommended by Lance et al., and C9S, C10S, RMSEAS, and CFIS proposed by McNeish and Hancock. The results in the present study showed that the false positive rates of C9S and C10S were high when the sample size was small or when factor loadings were low, although their sensitivity was negligibly impacted by the size of factor loadings. The higher sensitivity of RSMEA-P, C9, and C10 was observed in conditions with more observed items with higher factor loadings. Empirical implications for applied researchers were discussed.

An introduction to model implied instrumental variables using two stage least squares (MIIV-2SLS) in structural equation models (SEMs).

Structural equation models (SEMs) are widely used to handle multiequation systems that involve latent variables, multiple indicators, and measurement error. Maximum likelihood (ML) and diagonally weighted least squares (DWLS) dominate the estimation of SEMs with continuous or categorical endogenous variables, respectively. When a model is correctly specified, ML and DWLS function well. But, in the face of incorrect structures or nonconvergence, their performance can seriously deteriorate. Model implied instrumental variable, two stage least squares (MIIV-2SLS) estimates and tests individual equations, is more robust to misspecifications, and is noniterative, thus avoiding nonconvergence. This article is an overview and tutorial on MIIV-2SLS. It reviews the six major steps in using MIIV-2SLS: (a) model specification; (b) model identification; (c) latent to observed (L2O) variable transformation; (d) finding MIIVs; (e) using 2SLS; and (f) tests of overidentified equations. Each step is illustrated using a running empirical example from Reisenzein's (1986) randomized experiment on helping behavior. We also explain and illustrate the analytic conditions under which an equation estimated with MIIV-2SLS is robust to structural misspecifications. We include additional sections on MIIV approaches using a covariance matrix and mean vector as data input, conducting multilevel SEM, analyzing categorical endogenous variables, causal inference, and extensions and applications. Online supplemental material illustrates input code for all examples and simulations using the R package MIIVsem. (PsycInfo Database Record (c) 2022 APA, all rights reserved).

Testing the Performance of Level-Specific Fit Evaluation in MCFA Models With Different Factor Structures Across Levels.

A Monte Carlo study was conducted to compare the performance of a level-specific (LS) fit evaluation with that of a simultaneous (SI) fit evaluation in multilevel confirmatory factor analysis (MCFA) models. We extended previous studies by examining their performance under MCFA models with different factor structures across levels. In addition, various design factors and interaction effects between intraclass correlation (ICC) and misspecification type (MT) on their performance were considered. The simulation results demonstrate that the LS outperformed the SI in detecting model misspecification at the between-group level even in the MCFA model with different factor structures across levels. Especially, the performance of LS fit indices depended on the ICC, group size (GS), or MT. More specifically, the results are as follows. First, the performance of root mean square error of approximation (RMSEA) was more promising in detecting misspecified between-level models as GS or ICC increased. Second, the effect of ICC on the performance of comparative fit index (CFI) or Tucker-Lewis index (TLI) depended on the MT. Third, the performance of standardized root mean squared residual (SRMR) improved as ICC increased and this pattern was more clear in structure misspecification than in measurement misspecification. Finally, the summary and implications of the results are discussed.

A unified model-implied instrumental variable approach for structural equation modeling with mixed variables

The model-implied instrumental variable (MIIV) estimator is an equation-by-equation estimator of structural equation models that is more robust to structural misspecifications than full information estimators. Previous studies have concentrated on endogenous variables that are all continuous (MIIV-2SLS) or all ordinal . We develop a unified MIIV approach that applies to a mixture of binary, ordinal, censored, or continuous endogenous observed variables. We include estimates of factor loadings, regression coefficients, variances, and covariances along with their asymptotic standard errors. In addition, we create new goodness of fit tests of the model and overidentification tests of single equations. Our simulation study shows that the proposed MIIV approach is more robust to structural misspecifications than diagonally weighted least squares (DWLS) and that both the goodness of fit model tests and the overidentification equations tests can detect structural misspecifications. We also find that the bias in asymptotic standard errors for the MIIV estimators of factor loadings and regression coefficients are often lower than the DWLS ones, though the differences are small in large samples. Our analysis shows that scaling indicators with low reliability can adversely affect the MIIV estimators. Also, using a small subset of MIIVs reduces small sample bias of coefficient estimates, but can lower the power of overidentification tests of equations.

Comparing estimators for latent interaction models under structural and distributional misspecifications.

Estimation methods for structural equation models with interactions of latent variables were compared in several studies. Yet none of these studies examined models that were structurally misspecified. Here, the model-implied instrumental variable 2-stage least square estimator (MIIV-2SLS; Bollen, 1995; Bollen & Paxton, 1998), the 2-stage method of moments estimator (2SMM; Wall & Amemiya, 2003), the nonlinear structural equation mixture model approach (NSEMM; Kelava, Nagengast, & Brandt, 2014), and the unconstrained product indicator approach (UPI; Marsh, Wen, & Hau, 2004) were compared in a Monte Carlo simulation. The design included structural misspecifications in the measurement model involving the scaling indicator or not, the size of the misspecification, normal and nonnormal data, the indicators' reliability, and sample size. For the structural misspecifications that did not involve the scaling indicator, we found that MIIV-2SLS' parameter estimates were less biased compared with 2SMM, NSEMM, and UPI. If the reliability was high, the RMSE for all approaches was very similar; for low reliability, MIIV-2SLS' RMSE became larger compared with the other approaches. If the structural misspecification involved the scaling indicator, all estimators were seriously biased, with the largest bias for MIIV-2SLS. In most scenarios, this bias was more severe for the linear effects than for the interaction effect. The RMSE for conditions with misspecified scaling indicators was smallest for 2SMM, especially for low reliability scenarios, but the overall magnitude of bias was such that we cannot recommend any of the estimators in this situation. Our article showed the damage done when researchers omit cross-loadings of the scaling indicator and the importance of giving more attention to these indicators particularly if the indicators' reliability is low. It also showed that no one estimator is superior to the others across all conditions. (PsycInfo Database Record (c) 2020 APA, all rights reserved).

When Good Loadings Go Bad: Robustness in Factor Analysis

ABSTRACT Structural misspecifications in factor analysis include using the wrong number of factors and omitting cross loadings or correlated errors. The impact of these errors on factor loading estimates is understudied. Factor loadings underlie our assessments of the validity and reliability of indicators. Thus knowing how structural misspecifications affect a factor loading is a key issue. This paper develops analytic conditions of when misspecifications affect Bollen’s (1996) model implied instrumental variable, two stage least squares (MIIV-2SLS) estimator of a factor loading. It shows that if an indicator equation is correctly specified, then correlated errors among other measures, mixing up causal indicators with reflective, omitting cross loadings, and omitting direct effects between indicators leave the MIIV-2SLS estimator of the factor loading unchanged. Alternatively, if the indicator or the scaling indicator equation is misspecified, then the loading is unlikely to be robust. The results are illustrated with hypothetical and empirical examples.

Bayesian multivariate regime-switching models and the impact of correlation structure misspecification in variable annuity pricing

ABSTRACTWe develop Bayesian multivariate regime-switching models for correlated assets, comparing three different ways to flexibly structure the correlation matrix. After developing the models, we examine their relative characteristics and performance, first in a straightforward asset simulation example, and later applied to a variable annuity product with guarantees. We find that the freedom allowed by the more flexible structures enables these models to more accurately reflect the actual asset dependence structure. We also show that the correlation structures inferred by the most commonly used (and simplest) model will result in significantly larger estimates of the cost of the annuity guarantees.

Two-step Estimation for Longitudinal Data When the Working Correlation Matrix is a Linear Combination of Some Known Matrices

The generalized estimating equations(GEE) approach is perhaps one of the most widely used methods for longitudinal data analysis. While the GEE method guarantees the consistency of its estimators under working correlation structure misspecification, the corresponding efficiency can be severely affected. In this paper, we propose a new two-step estimation method in which the correlation matrix is assumed to be a linear combination of some known working matrices. Asymptotic properties of the new estimators are developed. Simulation studies are conducted to examine the performance of the proposed estimators. We illustrate the methodology with an epileptic data set.

A Limited Information Estimator for Dynamic Factor Models

Structural equation modeling (SEM) is an increasingly popular method for examining multivariate time series data. As in cross-sectional data analysis, structural misspecification of time series models is inevitable, and further complicated by the fact that errors occur in both the time series and measurement components of the model. In this article, we introduce a new limited information estimator and local fit diagnostic for dynamic factor models within the SEM framework. We demonstrate the implementation of this estimator and examine its performance under both correct and incorrect model specifications via a small simulation study. The estimates from this estimator are compared to those from the most common system-wide estimators and are found to be more robust to the structural misspecifications considered.

Model-Based Conditional Weighted Residuals Analysis for Structural Model Assessment

Nonlinear mixed effects models are widely used to describe longitudinal data to improve the efficiency of drug development process or increase the understanding of the studied disease. In such settings, the appropriateness of the modeling assumptions is critical in order to draw correct conclusions and must be carefully assessed for any substantial violations. Here, we propose a new method for structure model assessment, based on assessment of bias in conditional weighted residuals (CWRES). We illustrate this method by assessing prediction bias in two integrated models for glucose homeostasis, the integrated glucose-insulin (IGI) model, and the integrated minimal model (IMM). One dataset was simulated from each model then analyzed with the two models. CWRES outputted from each model fitting were modeled to capture systematic trends in CWRES as well as the magnitude of structural model misspecifications in terms of difference in objective function values (ΔOFVBias). The estimates of CWRES bias were used to calculate the corresponding bias in conditional predictions by the inversion of first-order conditional estimation method’s covariance equation. Time, glucose, and insulin concentration predictions were the investigated independent variables. The new method identified correctly the bias in glucose sub-model of the integrated minimal model (IMM), when this bias occurred, and calculated the absolute and proportional magnitude of the resulting bias. CWRES bias versus the independent variables agreed well with the true trends of misspecification. This method is fast easily automated diagnostic tool for model development/evaluation process, and it is already implemented as part of the Perl-speaks-NONMEM software.

Problems with Rationales for Parceling that Fail to Consider Parcel-Allocation Variability

In structural equation modeling applications, parcels—averages or sums of subsets of item scores—are often used as indicators of latent constructs. Parcel-allocation variability (PAV) is variability in results that arises within sample across alternative item-to-parcel allocations. PAV can manifest in all results of a parcel-level model (e.g., model fit, parameter estimates, standard errors, and inferential decisions). It is a source of uncertainty in parcel-level model results that can be investigated, reported, and accounted for. Failing to do so raises representativeness and replicability concerns. However, in recent methodological literature (Cole, Perkins, & Zelkowitz, 2016; Little, Rhemtulla, Gibson, & Shoemann, 2013; Marsh, Ludtke, Nagengast, Morin, & von Davier, 2013; Rhemtulla, 2016) parceling has been justified and recommended in several situations without quantifying or accounting for PAV. In this article, we explain and demonstrate problems with these rationales. Overall, we find that: (1) using a purposive parceling algorithm for a multidimensional construct does not avoid PAV; (2) passing a test of unidimensionality of the item-level model need not avoid PAV; and (3) a desire to improve power for detecting structural misspecification does not warrant parceling without addressing PAV; we show how to simultaneously avoid PAV and obtain even higher power by comparing item-level models differing in structural constraints. Implications for practice are discussed.

Impact of error structure misspecification when testing measurement invariance and latent-factor mean difference using MIMIC and multiple-group confirmatory factor analysis.

When multiple groups are compared, the error variance-covariance structure is not always invariant between groups. In this study we investigated the impacts of misspecified error structures on testing measurement invariance and the latent-factor mean difference between groups. A Monte Carlo study was conducted to examine how measurement invariance and latent mean difference tests were affected when heterogeneous error structures were misspecified as being invariant across groups. Multiple-group confirmatory factor analysis (MGCFA) and the multiple-indicator multiple-causes model (MIMIC) were employed in the present study. The rejection rates of both metric and strict invariance in measurement invariance testing, as well as the estimation accuracy and statistical inference of the factor mean difference, were investigated under error structure misspecification. In addition, sensitivity of the model fit indices to error structure misspecification was examined. Overall, misspecification of the error structure affected testing for metric but not scalar invariance. Metric invariance was often rejected, especially when error covariance in one group was ignored. In contrast, MGCFA and MIMIC performed comparatively well at detecting latent-factor mean differences between groups, with acceptable power and well-controlled Type I errors. The practical implications of these findings are discussed, as well as recommendations.

Model Implied Instrumental Variables (MIIVs): An Alternative Orientation to Structural Equation Modeling

Few dispute that our models are approximations to reality. Yet when it comes to structural equation models (SEMs), we use estimators that assume true models (e.g. maximum likelihood) and that can create biased estimates when the model is inexact. This article presents an overview of the Model Implied Instrumental Variable (MIIV) approach to SEMs from Bollen (1996). The MIIV estimator using Two Stage Least Squares (2SLS), MIIV-2SLS, has greater robustness to structural misspecifications than system wide estimators. In addition, the MIIV-2SLS estimator is asymptotically distribution free. Furthermore, MIIV-2SLS has equation-based overidentification tests that can help pinpoint misspecifications. Beyond these features, the MIIV approach has other desirable qualities. MIIV methods apply to higher order factor analyses, categorical measures, growth curve models, dynamic factor analysis, and nonlinear latent variables. Finally, MIIV-2SLS permits researchers to estimate and test only the latent variable model or any other subset of equations. In addition, other MIIV estimators beyond 2SLS are available. Despite these promising features, research is needed to better understand its performance under a variety of conditions that represent empirical applications. Empirical and simulation examples in the article illustrate the MIIV orientation to SEMs and highlight an R package MIIVsem that implements MIIV-2SLS.

Robustness Conditions for MIIV-2SLS When the Latent Variable or Measurement Model is Structurally Misspecified

Most researchers acknowledge that virtually all structural equation models (SEMs) are approximations due to violating distributional assumptions and structural misspecifications. There is a large literature on the unmet distributional assumptions, but much less on structural misspecifications. In this paper, we examine the robustness to structural misspecification of the model implied instrumental variable, two-stage least square (MIIV-2SLS) estimator of SEMs. We introduce two types of robustness: robust-unchanged and robust-consistent. We develop new robustness analytic conditions for MIIV-2SLS and illustrate these with hypothetical models, simulated data, and an empirical example. Our conditions enable a researcher to know whether, for example, a structural misspecification in the latent variable model influences the MIIV-2SLS estimator for measurement model equations and vice versa. Similarly, we establish robustness conditions for correlated errors. The new robustness conditions provide guidance on the types of structural misspecifications that affect parameter estimates and they assist in diagnosing the source of detected problems with MIIVs.

Conditional mix-GEE models for longitudinal data with unspecified random-effects distributions

ABSTRACTIn the longitudinal studies, the mixture generalized estimation equation (mix-GEE) was proposed to improve the efficiency of the fixed-effects estimator for addressing the working correlation structure misspecification. When the subject-specific effect is one of interests, mixed-effects models were widely used to analyze longitudinal data. However, most of the existing approaches assume a normal distribution for the random effects, and this could affect the efficiency of the fixed-effects estimator. In this article, a conditional mixture generalized estimating equation (cmix-GEE) approach based on the advantage of mix-GEE and conditional quadratic inference function (CQIF) method is developed. The advantage of our new approach is that it does not require the normality assumption for random effects and can accommodate the serial correlation between observations within the same cluster. The feature of our proposed approach is that the estimators of the regression parameters are more efficient than CQIF even if the working correlation structure is not correctly specified. In addition, according to the estimates of some mixture proportions, the true working correlation matrix can be identified. We establish the asymptotic results for the fixed-effects parameter estimators. Simulation studies were conducted to evaluate our proposed method.

A Simulation Study of Polychoric Instrumental Variable Estimation in Structural Equation Models

Data collected from questionnaires are often in ordinal scale. Unweighted least squares (ULS), diagonally weighted least squares (DWLS) and normal-theory maximum likelihood (ML) are commonly used methods to fit structural equation models. Consistency of these estimators demands no structural misspecification. In this article, we conduct a simulation study to compare the equation-by-equation polychoric instrumental variable (PIV) estimation with ULS, DWLS, and ML. Accuracy of PIV for the correctly specified model and robustness of PIV for misspecified models are investigated through a confirmatory factor analysis (CFA) model and a structural equation model with ordinal indicators. The effects of sample size and nonnormality of the underlying continuous variables are also examined. The simulation results show that PIV produces robust factor loading estimates in the CFA model and in structural equation models. PIV also produces robust path coefficient estimates in the model where valid instruments are used. However, robustness highly depends on the validity of instruments.

Reducing hydrologic model uncertainty in monthly streamflow predictions using multimodel combination

Model errors are inevitable in any prediction exercise. One approach that is currently gaining attention in reducing model errors is by combining multiple models to develop improved predictions. The rationale behind this approach primarily lies on the premise that optimal weights could be derived for each model so that the developed multimodel predictions will result in improved predictions. A new dynamic approach (MM‐1) to combine multiple hydrological models by evaluating their performance/skill contingent on the predictor state is proposed. We combine two hydrological models, “abcd” model and variable infiltration capacity (VIC) model, to develop multimodel streamflow predictions. To quantify precisely under what conditions the multimodel combination results in improved predictions, we compare multimodel scheme MM‐1 with optimal model combination scheme (MM‐O) by employing them in predicting the streamflow generated from a known hydrologic model (abcd model orVICmodel) with heteroscedastic error variance as well as from a hydrologic model that exhibits different structure than that of the candidate models (i.e., “abcd” model or VIC model). Results from the study show that streamflow estimated from single models performed better than multimodels under almost no measurement error. However, under increased measurement errors and model structural misspecification, both multimodel schemes (MM‐1 and MM‐O) consistently performed better than the single model prediction. Overall, MM‐1 performs better than MM‐O in predicting the monthly flow values as well as in predicting extreme monthly flows. Comparison of the weights obtained from each candidate model reveals that as measurement errors increase, MM‐1 assigns weights equally for all the models, whereas MM‐O assigns higher weights for always the best‐performing candidate model under the calibration period. Applying the multimodel algorithms for predicting streamflows over four different sites revealed that MM‐1 performs better than all single models and optimal model combination scheme, MM‐O, in predicting the monthly flows as well as the flows during wetter months.